Branko Bijeljic

Mixing, spreading and reaction in heterogeneous media: A brief review

Geological media exhibit heterogeneities in their hydraulic and chemical properties, which can lead to enhanced spreading and mixing of the transported species and induce an effective reaction behavior that is different from the one for a homogeneous medium. Chemical heterogeneities such as spatially varying adsorption properties and specific reactive surface areas can act directly on the chemical reaction dynamics and lead to different effective reaction laws. Physical heterogeneities affect mixing-limited chemical reactions in an indirect way by their impact on spreading and mixing of dissolved species. To understand and model large-scale reactive transport the interactions of these coupled processes need to be understood and quantified. This paper provides a brief review on approaches of non-reactive and reactive transport modeling in geological media.

Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method

We present a stable numerical scheme for modelling multiphase flow in porous media, where the characteristic size of the flow domain is of the order of microns to millimetres. The numerical method is developed for efficient modelling of multiphase flow in porous media with complex interface motion and irregular solid boundaries. The Navier-Stokes equations are discretised using a finite volume approach, while the volume-of-fluid method is used to capture the location of interfaces. Capillary forces are computed using a semi-sharp surface force model, in which the transition area for capillary pressure is effectively limited to one grid block. This new formulation along with two new filtering methods, developed for correcting capillary forces, permits simulations at very low capillary numbers and avoids non-physical velocities. Capillary forces are implemented using a semi-implicit formulation, which allows larger time step sizes at low capillary numbers. We verify the accuracy and stability of the numerical method on several test cases, which indicate the potential of the method to predict multiphase flow processes.

Pore‐scale modeling of longitudinal dispersion

[1] We study macroscopic (centimeter scale) dispersion using pore-scale network simulation. A Lagrangian-based transport model incorporating flow and diffusion is applied in a diamond lattice of throats with square cross section whose radius distribution is the same as computed for Berea sandstone. We use physically consistent rules using a combination of stream-tube routing and ideal mixing to transport particles across pore junctions. The influence of both heterogeneity and high Peclet numbers results in asymptotic behavior only being seen after movement through many throats. A comprehensive comparative study of longitudinal dispersion with experiments in consolidated and unconsolidated media indicates that the model can quantitatively predict the asymptotic macroscopic dispersion coefficient over a broad range of Peclet numbers, 0 400.

Insights into non‐Fickian solute transport in carbonates

[1] We study and explain the origin of early breakthrough and long tailing plume behavior by simulating solute transport through 3-D X-ray images of six different carbonate rock samples, representing geological media with a high degree of pore-scale complexity. A Stokes solver is employed to compute the flow field, and the particles are then transported along streamlines to represent advection, while the random walk method is used to model diffusion. We compute the propagators (concentration versus displacement) for a range of Peclet numbers (Pe) and relate it to the velocity distribution obtained directly on the images. There is a very wide distribution of velocity that quantifies the impact of pore structure on transport. In samples with a relatively narrow spread of velocities, transport is characterized by a small immobile concentration peak, representing essentially stagnant portions of the pore space, and a dominant secondary peak of mobile solute moving at approximately the average flow speed. On the other hand, in carbonates with a wider velocity distribution, there is a significant immobile peak concentration and an elongated tail of moving fluid. An increase in Pe, decreasing the relative impact of diffusion, leads to the faster formation of secondary mobile peak(s). This behavior indicates highly anomalous transport. The implications for modeling field-scale transport are discussed. Citation: Bijeljic, B., P. Mostaghimi, and M. J. Blunt (2013), Insights into non-Fickian solute transport in carbonates, Water Resour. Res., 49, 2714–2728, doi:10.1002/wrcr.20238.

Pore‐by‐pore capillary pressure measurements using X‐ray microtomography at reservoir conditions: Curvature, snap‐off, and remobilization of residual CO2

X-ray microtomography was used to image the shape and size of residual ganglia of supercritical CO2 at resolutions of 3.5 and 2 μm and at representative subsurface conditions of temperature and pressure. The capillary pressure for each ganglion was found by measuring the curvature of the CO2-brine interface, while the pore structure was parameterized using distance maps of the pore space. The formation of the residual clusters by snap-off was examined by comparing the ganglion capillary pressure to local pore topography. The capillary pressure was found to be inversely proportional to the radius of the largest restriction (throat) surrounding the ganglion, which validates the imbibition mechanisms used in pore-network modeling. The potential mobilization of residual ganglia was assessed using a reformulation of both the capillary (Ncmacro) and Bond numbers (Nbmacro), rigorously based on a balance of pore-scale forces, with the majority of ganglia remobilized at Ncmacro around 1. Buoyancy forces were found to be small in this system (Nbmacro << 1), meaning the gravitational remobilization of CO2 after residual trapping would be extremely difficult.

Simulation of Flow and Dispersion on Pore-Space Images

We simulate flow and transport directly on pore-space images obtained from a microcomputed-tomography (micro-CT) scan of rock cores. An efficient Stokes solver is used to simulate lowReynolds-number flows. The flow simulator uses a finite-difference method along with a standard predictor/corrector procedure to decouple pressure and velocity. An algebraic multigrid technique solves the linear systems of equations. We then predict permeability, and the results are compared with lattice-Boltzmannmethod (LBM) numerical results and available experimental data. For solute transport, we apply a streamline-based algorithm that is similar to the Pollock algorithm common in field-scale reservoir simulation, but which uses a novel semianalytic formulation near solid boundaries to capture, with subgrid resolution, the variation in velocity near the grains. A random-walk method accounts for molecular diffusion. The streamline-based algorithm is validated by comparison with published results for Taylor-Aris dispersion in a single capillary with a square cross section. We then predict accurately the available experimental data in the literature for the longitudinal dispersion coefficient for a range of Péclet numbers (10 to 10). We introduce a characteristic length on the basis of the ratio of volume to pore/grain surface area that can be used for consolidated porous media to calculate the Péclet number.

Measurement of Nonwetting-Phase Trapping in Sandpacks

Summary We measure the trapped nonwetting-phase saturation as a function of initial saturation in sandpacks. The application of the work is for carbon dioxide (CO2) storage in aquifers, where capillary trapping is a rapid and effective mechanism to render the injected fluid immobile: The CO2 is injected into the formation followed by chase-brine injection or natural groundwater flow that displaces and traps it. Current models to predict the amount of trapping are based on experiments in consolidated media; while CO2 is likely to be injected at depths greater than approximately 800 m to render it supercritical, it may be injected into formations that tend to have a higher porosity and permeability than deep oilfield rocks. We use analog fluids—water and refined oil—at ambient conditions. The initial conditions are established by injecting oil into vertical or horizontal sandpacks 0.6 m long at different flow rates and then allowing the oil to migrate under gravity. The packs are then flooded with water. The columns are sliced, and the residual saturation is measured with great accuracy and sensitivity by gas chromatography (GC). This method allows low saturations to be measured reliably. The trapped saturation initially rises linearly with initial saturation to a value of approximately 0.13, followed by a constant residual as the initial saturation increases further. This behavior is not predicted by the traditional Land (1968) model but is physically consistent with poorly consolidated media where most of the larger pores can be invaded easily at relatively low saturation and there is, overall, relatively little trapping. The best match to our experimental data is achieved with the Aissaoui (1983) and the Spiteri et al. (2008) trapping models.

Dynamic three-dimensional pore-scale imaging of reaction in a carbonate at reservoir conditions

Quantifying CO2 transport and average effective reaction rates in the subsurface is essential to assess the risks associated with underground carbon capture and storage. We use X-ray microtomography to investigate dynamic pore structure evolution in situ at temperatures and pressures representative of underground reservoirs and aquifers. A 4 mm diameter Ketton carbonate core is injected with CO2-saturated brine at 50 °C and 10 MPa while tomographic images are taken at 15 min intervals with a 3.8 μm spatial resolution over a period of 2(1/2) h. An approximate doubling of porosity with only a 3.6% increase in surface area to volume ratio is measured from the images. Pore-scale direct simulation and network modeling on the images quantify an order of magnitude increase in permeability and an appreciable alteration of the velocity field. We study the uniform reaction regime, with dissolution throughout the core. However, at the pore scale, we see variations in the degree of dissolution with an overall reaction rate which is approximately 14 times lower than estimated from batch measurements. This work implies that in heterogeneous rocks, pore-scale transport of reactants limits dissolution and can reduce the average effective reaction rate by an order of magnitude.

Pore-scale intermittent velocity structure underpinning anomalous transport through 3-D porous media

We study the nature of non-Fickian particle transport in 3-D porous media by simulating fluid flow in the intricate pore space of real rock. We solve the full Navier-Stokes equations at the same resolution as the 3-D micro-CT (computed tomography) image of the rock sample and simulate particle transport along the streamlines of the velocity field. We find that transport at the pore scale is markedly anomalous: longitudinal spreading is superdiffusive, while transverse spreading is subdiffusive. We demonstrate that this anomalous behavior originates from the intermittent structure of the velocity field at the pore scale, which in turn emanates from the interplay between velocity heterogeneity and velocity correlation. Finally, we propose a continuous time random walk model that honors this intermittent structure at the pore scale and captures the anomalous 3-D transport behavior at the macroscale.

Numerical Modelling of Sub-pore Scale Events in Two-Phase Flow Through Porous Media

We use a new volume-of-fluid based finite-volume method to model two-phase flow through simple pore geometries and study the mechanisms controlling two-phase flow at the pore scale. The numerical model is used to study layer flow and snap-off, and investigate the effect of geometry and flow rate on trapping and mobilization of the disconnected ganglia. Furthermore, a new variable, the capillary field, is introduced to characterize the capillary force under dynamic situations, and a force-balance concept is presented to relate flow rates to pore-scale forces—dynamic pressure gradient and the capillary field. This description of the flow has the potential to be used in pore-network models to study the effect of pore-scale structures on the flow at larger scales. As an illustration of the applicability of this concept, we use the relations obtained from the numerical simulations to predict the threshold capillary number for blob mobilization during imbibition and show that this information can be used to reproduce the direct numerical simulation results accurately.